How can a Method from a startup Script be invoked?
To answer your question:

Operator types are automatically added to bpy.types on registration.

This is how a method on an operator your register can be called.

bpy.types.OBJECT_OT_util.dump(bpy.context.object)


Note that it is very rare to ever access operator types like this, its more of an internal implementation detail.

normally you would access the operator via bpy.ops.object_utils() but this is only for calling the operator and won't expose its original class.

Suggested alternative

In this case I would suggest not to having dump() as an operator class method at all.
If you have a function that can run outside of the operator, you're better off having it as a utility function in a module other scripts can import and use, then have the both the operator and external scripts use it.

my_module.py:

def dump(...):
   ... do stuff ...

class MyOperator(Operator):
    ...
    def execute(self, context):
        dump()


So now any script can do:

import my_module
my_module.dump()


This way we bypass Blender/Python API internals where its not helpful and let Python modules interact as they would typically  (outside of Blender).